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# An Optimal Algorithm for Strongly Convex Minimization under Affine Constraints

*Proceedings of The 25th International Conference on Artificial Intelligence and Statistics*, PMLR 151:4482-4498, 2022.

#### Abstract

Optimization problems under affine constraints appear in various areas of machine learning. We consider the task of minimizing a smooth strongly convex function F(x) under the affine constraint Kx = b, with an oracle providing evaluations of the gradient of F and multiplications by K and its transpose. We provide lower bounds on the number of gradient computations and matrix multiplications to achieve a given accuracy. Then we propose an accelerated primal-dual algorithm achieving these lower bounds. Our algorithm is the first optimal algorithm for this class of problems.